\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 279, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/279\hfil Nonlocal time-delay wave equation]
{Well-posedness and exponential stability for a wave equation with
 nonlocal time-delay condition}

\author[C. A. Raposo, H. Nguyen, J. O. Ribeiro, V. Barros \hfil EJDE-2017/279\hfilneg]
{Carlos A. Raposo, Hoang Nguyen, Joilson O. Ribeiro, Vanessa Barros}

\address{Carlos Alberto Raposo \newline
Departamento de Matem\'atica e Estat\'istica,
 Universidade Federal de S\~ao Jo\~ao del-Rei, Brazil.\newline
Instituto de Matem\'atica, Universidade Federal da Bahia, Brazil}
\email{raposo@ufsj.edu.br}

\address{Huy Hoang Nguyen \newline
Instituto de Matem\'atica and Campus de Xer\'em,
Universidade Federal do Rio de Janeiro, Brazil. \newline
Laboratoire de Math\'ematiques et de leurs Applications
(LMAP/UMR CNRS 5142), Bat. IPRA,
 Avenue de l'Universit\'e, F-64013, France}
\email{nguyen@im.ufrj.br}

\address{Joilson Oliveira Ribeiro \newline
Instituto de Matem\'atica,
Universidade Federal da Bahia, Brazil}
\email{joilsonor@ufba.br}

\address{Vanessa Barros de Oliveira \newline
Instituto de Matem\'atica,
 Universidade Federal da Bahia, Brazil}
\email{vbarrosoliveira@gmail.com}

\dedicatory{Communicated by Ludmila S. Pulkina}

\thanks{Submitted July 24, 2017. Published November 10, 2017.}
\subjclass[2010]{35L05,  35B35, 35L51}
\keywords{Well-posedness; exponential stability; wave equation; semigroup}

\begin{abstract}
 Well-posedness and exponential stability of nonlocal time-delayed of
 a wave equation with a integral conditions of the 1st kind forms the center
 of this work.  Through semigroup theory we prove the well-posedness by
 the Hille-Yosida theorem and the exponential stability  exploring  the
 dissipative properties of the linear operator associated to damped model
 using the Gearhart-Huang-Pruss theorem.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

\section{Introduction}

Let $\Omega = (0,1)$  be an interval in $\mathbb{R}$,
$(x,t) \in \Omega \times (0,\infty) $ and $a, b$ be positive constants.
We denote by $u=u(x,t)$  the small transversal displacements of $x$
at the time $t$. The wave equation with frictional damping is modeled by
\begin{equation} \label{e1.1}
u_{tt} - au_{xx} + bu_t = 0.
\end{equation}
Nonlocal time-delayed wave equation forms the center of this work.
One of the first approach for a model with delay was given by Ludwig
Eduard Boltzmann (1844-1906) who studied retarded elasticity effects.
Charles \'Emile Picard (1856-1941) took the view that the past states are
important for a realistic modelling although the Newtonian tradition
claimed the opposite. The need for delays was emphasized both by Lotka
and by Volterra independently of each other, Alfred J. Lotka (1880-1949)
in the United States and Vito Volterra (1860-1940) in Italy.
They introduced the Lotka-Volterra equations, also known as the predator-prey
equations.  Andrey Nikolaevich Kolmogorov (1903-1987) introduced the model
which is a more general framework that can model the dynamics of ecological
systems with predator-prey interactions, competition, disease, and mutualism.
Anatoliy Myshkis (1920-2009) gave the first correct mathematical formulation
and introduced a general class of equations with delayed arguments and laid
the foundation for a general theory of linear systems.
In fact, time delays so often arise in many physical, chemical, biological and
economical phenomena (see \cite{Suh} and references therein).
Whenever that material, information or energy is physically transmitted
from one place to another, there is a delay associated with the transmission
and in this direction the delay is the property of a physical system by which
the response to an applied restoring force is delayed in its transmission
effect  (see \cite{Shinskey}). Unluckily, the delay can becomes a source
of instability. In the work \cite{Datko}, a small delay in a boundary
control could turn the well-behaved hyperbolic system into a wild one was shown.
See for example \cite{ Datko2, Guesmia, Nicaise_Pignotti_2, Nicaise_Pignotti, Xu}
where an arbitrarily small delay may destabilize a system that is uniformly
asymptotically stable in the absence of delay unless additional control terms
have been used.

The history of nonlocal problems with integral conditions  for partial
differential equations is recent and goes back to \cite{Cannon}.
In \cite{Bazant}, a review of the progress in the nonlocal models with
integral type was given with many discussions related to  physical
justifications, advantages, and numerical applications. For nonlocal problem
for a hyperbolic equation with integral conditions
of the 1st kind we cite \cite{Pulkina4}. We define the nonlocal time-delay
integral of the 1st kind  condition by
\begin{equation}
\int_0^c F(s)u_t(x, t-s) ds. \label{NLC}
\end{equation}
This kind of condition \eqref{NLC} is called nonlocal because the integral
is not a pointwise relation. The nonlocal terms provoke some mathematical
difficulties which makes the study of such a problem particularly interesting.
For  the  last  several  decades,  various  types  of  equations  have
been  employed  as some mathematical models describing physical, chemical,
biological and ecological systems.  See for example the nonlocal
reaction-diffusion system given in \cite{Raposo} and reference therein.
In \cite{Kozhanov} the authors  considered a nonlocal problem for a
 hyperbolic equation in $n$ space variables with a different integral condition.
 For mixed problems with nonlocal integral conditions in one-dimensional
hyperbolic equation, we cite the works
\cite{Pulkina, Bouziani, Gordeziani,  Beilin2,  Pulkina2, Pulkina3}.
For a nonlocal problem for wave equation with integral condition on a cylinder,
we cite \cite{Beilin} where the existence of a generalized solution
by Galerkin procedure was proved. In  \cite{MC},  Cavalcanti et al recently
considered a nonlinear wave equation with a degenerate and nonlocal damping term.
They proved the exponential stability borrowing some ideas  in
\cite{Dehman, Dehman2}. The generelized solution for a mixed nonlocal system
of wave equation was given in \cite{Ducival}. Stability for coupled wave
system has been considered in several works, for example in
\cite{Komornik, Assila1, Assila2, Raposo3, Najafi} among others.

For $\Omega \subset \mathbb{R}^n$ a open bounded domain with a smooth boundary,
in \cite{Nicaise_Pignotti_2} was considered the system with internal feedback
\begin{equation*}
u_{tt} - \Delta u +\mu_0u_t + \int_{\tau_1}^{\tau_2} a(s)\mu(s)u_t(x, t-s) ds
= 0, \quad (x,t) \in \Omega \times (0,\infty) %\label{P1}
\end{equation*}
and  assuming
$$
\mu_0 \geq \|a\|_{\infty}\int_{\tau_1}^{\tau_2}\mu(s)\,ds
$$
was proved the exponential decay of solution by Energy Method,
that consists in use of suitable multiplies to construct a
Lyapunov functional, equivalent to energy functional, that decay exponentially.

Recently, using the Energy Method,  Pignotti \cite{C_Pignotti} studied the
asymptotic and exponential stability results under suitable conditions
for the abstract model of second-order evolution equations \eqref{C_Nicaise_model}
\begin{equation}
u_{tt} + A u +\mu_0u_t - \int_{0}^{\infty}\mu(s)\mu(s)Au(x, t-s) ds
= 0, \quad (x,t) \in \Omega \times (0,\infty), \label{C_Nicaise_model}
\end{equation}
where a viscoelastic damping takes the place of the standard frictional
term and the delay feedback is intermittent on–off in time, being
$A: D(A)\to H $ a positive self-adjoint operator with compact inverse
in a real Hilbert space $H$.

In this work we use a different approach by semigroup technique and we
prove the well-posedness and exponential stability for a wave equation
with frictional damping and nonlocal time-delayed condition given by
\begin{gather}
u_{tt} - au_{xx} + bu_t + \int_0^c F(s)u_t(x, t-s) ds
= 0, \quad (x,t) \in \Omega \times (0,\infty), \label{P1}\\
u(x,0)= u_0(x),\quad  x \in \Omega, \\
u_t(x,0) = u_1(x),\quad  x \in \Omega,  \\
u_t(x,-s) = f_0(x,-s), \quad  x \in \Omega,\; s \in (0,c), \label{P4}
\end{gather}
and satisfying the Dirichlet boundary conditions
\begin{equation} \label{Dic1}
u(0,t) = u(1,t)= 0,\quad  t>0.
\end{equation}
Here the initial data $u_0(x) \in H^1_0(0,1)$, $u_1(x) \in L^2(0,1)$, $f_0(x,s)$
belongs to suitable space and
$$
\int_0^c F(s) ds < b.
$$
We use the Sobolev spaces and semigroup theory with its properties
as in \cite{Adams, Pazy, ZL}.
This article is organized as follows. In section 2, we present some notation
 and assumptions needed to establish the well-posedness.
In section 3, we prove  the exponential stability using the
Gearhart-Huang-Pruss theorem (see \cite{Gearhart,Huang,Pruss}).

\section{Well-posedness}

As in Nicaise and Pignotti \cite{Nicaise_Pignotti_2} we introduce the new variable
$$
z(x,\rho,t,s) = u_t(x,t -\rho s),\quad
 (x,\rho)\in Q=\Omega \times \Omega,\quad  t>0,\; s \in (0,c).
$$
The new variable $z$ satisfies
\begin{equation}
sz_t(x,\rho,t,s) + z_\rho(x,\rho,t,s) = 0. \label{e2.1}
\end{equation}
Moreover, using the approach as in \cite{Nicaise_Pignotti},
 the  equation
\begin{equation}
\lambda sz(x,\rho,t,s) + z_\rho(x,\rho,t,s) = f, \quad
\text{with } \lambda > 0, \; f \in L^2(Q \times (0,c)) \label{e2.2}
\end{equation}
has a unique solution
\begin{equation}
z(x,\rho,s)  =  z(x,0,s) e^{-\lambda \rho s}
+ s   e^{\lambda \rho s}\int_0^\rho e^{\lambda \sigma s}f(x,\sigma,s) d\sigma. \label{e2.3}
\end{equation}
Consequently, problem \eqref{P1}-\eqref{P4} is equivalent to
\begin{gather}
u_{tt} - au_{xx} + bu_t + \int_0^c F(s)z(x,1,t,s) ds
= 0,  \quad  (x,t) \in \Omega \times (0,\infty),\label{P5}\\
sz_t(x,\rho,t,s) + z_\rho(x,\rho,t,s) = 0, (x,\rho) \in Q, t>0, s\in (0,c),  \\
u(x,0) = u_0(x),\quad x \in \Omega,  \\
u_t(x,0) = u_1(x),\quad x \in \Omega,  \\
z(x,1,s) = f_0(x,-s), \quad  x \in \Omega,\; s \in (0,c), \label{P9}
\end{gather}
with the Dirichlet boundary condition \eqref{Dic1} and $z(x,\rho,t,s)=0$
on the boundary.
Defining $U = (u,v,z)^T $, $ v= u_t$, we formally get that $U$ satisfies
the Cauchy problem
\begin{equation}
\begin{gathered}
U_t=\mathcal{A}U \quad  t>0,\\
U(0)=U_0 =(u_0,v_0,f_0)^T,
\end{gathered} \label{e2.9}
\end{equation}
where the operator $\mathcal{A}$ is defined as
\begin{equation*}
\mathcal{A}U=\begin{bmatrix}
v\\
au_{xx} - bv - \int_0^c F(s)z(x,1,t,s) ds\\
-s^{-1}z_\rho(x,\rho,t,s)
\end{bmatrix}. %\label{1:15}
\end{equation*}
We introduce the energy space
$$
\mathcal{H}= H_0^1(\Omega)\times L^2(\Omega)\times L^2(Q\times (0,c))
$$
equipped with the inner product
\begin{equation*}
\langle U,\bar{U}\rangle_{\mathcal{H}}
= \int_\Omega ( a u_x \bar{u}_x +  v \bar{v})\,dx
+ \int_{\Omega}\int_{\Omega}
\Big[ \int_0^c s F(s)z(x,\rho,s)\bar{z}(x,\rho,s)  ds \Big]d\rho \,dx.
\end{equation*}
The domain of $\mathcal{A}$ is
$$
D(\mathcal{A}) = H^2(\Omega)\cap H_0^1(\Omega)\times H_0^1(\Omega)
\times L^2(Q\times (0,c)).
$$
Clearly, $D(\mathcal{A})$ is dense in $\mathcal{H}$ and independent of time $t>0$.
Next, we prove that the operator $\mathcal{A}$ is dissipative.

\begin{proposition} \label{prop2.1}
For $U=(u,v,z) \in D(\mathcal{A})$ we have
\begin{equation}
\langle \mathcal{A}U,U\rangle_{\mathcal{H} }
\leq -( b - \int_0^c F(s)ds)\int_\Omega v^2 \,dx.  \label{e2.10}
\end{equation}
\end{proposition}

\begin{proof}
We have
\begin{align*}
\langle \mathcal{A}U,U \rangle_{\mathcal{H} }
&= \int_\Omega \Big\{a v_x u_x +  \Big[ au_{xx} -bv - \int_0^c F(s)z(x,1,s)  ds
\Big]v\Big\}\,dx \\
&\quad - \int_{\Omega}\int_{\Omega}
\Big[ \int_0^c  F(s)z_\rho(x,\rho,s)z(x,\rho,s)  ds \Big]d\rho \,dx. %\label{17}
\end{align*}
\begin{align*}
\langle \mathcal{A}U,U \rangle_{\mathcal{H} }
&= a\int_\Omega v_x u_x \,dx +  a\int_\Omega u_{xx}v \,dx - b\int_\Omega v^2\,dx\\
& \quad - \int_\Omega\int_0^c F(s)z(x,1,s)v  \,ds\,dx \\
& \quad - \int_{\Omega}\int_{\Omega}
\Big[ \int_0^c  F(s)z_\rho(x,\rho,s)z(x,\rho,s)  ds \Big]d\rho \,dx. %\label{17}
\end{align*}
Integrating by parts in $\Omega$,
\begin{align*}
\langle \mathcal{A}U,U \rangle_{\mathcal{H} }
&= - b\int_\Omega v^2\,dx \\
&\quad -  \int_\Omega\int_0^c F(s)\,z(x,1,s)\,v \, ds\,dx  \\
&\quad - \int_{\Omega}\int_{\Omega}
\Big[ \int_0^c  F(s)z_\rho(x,\rho,s)z(x,\rho,s)  ds \Big]d\rho \,dx. %\label{17}
\end{align*}
Using $z(x,0,s) = u_t(x,t)=v$ note that
\begin{align*}
&\int_{\Omega}\int_{\Omega}
\Big[\int_0^c  F(s)z_\rho(x,\rho,s)z(x,\rho,s)  ds \Big]d\rho \,dx \\
&= \int_{\Omega}\int_0^c  \Big[\int_{\Omega}F(s)\frac{1}{2}
 \frac{d}{d\rho}|z(x,\rho,s)|^2  d\rho  \Big]ds \,dx  \\
&= \frac{1}{2}\int_{\Omega}\int_0^c F(s)|z(x,1,s)|^2ds\,dx
 - \frac{1}{2}\int_{\Omega}\int_0^cF(s)v^2ds\,dx.
\end{align*}
Then we have
\begin{align*}
\langle \mathcal{A}U,U \rangle_{\mathcal{H} }
&= - b\int_\Omega v^2\,dx
 - \int_\Omega\int_0^c F(s)\,z(x,1,s)\,v \, ds\,dx  \\
&\quad -  \frac{1}{2}\int_{\Omega}\int_0^c F(s)|z(x,1,s)|^2ds\,dx
 +\frac{1}{2}\int_0^c F(s)ds\int_\Omega v^2 \,dx.  %\label{17}
\end{align*}
Applying Young's inequality,
\begin{align*}
\langle \mathcal{A}U,U \rangle_{\mathcal{H}} 
&= - b\int_\Omega v^2\,dx 
 + \frac{1}{2}\int_\Omega\int_0^c F(s)|z(x,1,s)|^2ds\,dx
 + \frac{1}{2}\int_0^c F(s)ds\int_\Omega v^2 \,dx  \\
&\quad -  \frac{1}{2}\int_{\Omega}\int_0^c F(s)|z(x,1,s)|^2ds\,dx
 +  \frac{1}{2}\int_0^c F(s)ds\int_\Omega v^2 \,dx. %\label{17}
\end{align*}
From where it follows that
\begin{equation*}
\langle \mathcal{A}U,U\rangle_{\mathcal{H} }
\leq -( b - \int_0^c F(s)ds)\int_\Omega v^2 \,dx.  %\label{dissipative}
\end{equation*}
\end{proof}

The well-posedness of \eqref{P5}-\eqref{P9} is ensured by the following theorem.

\begin{theorem} \label{thm2.2}
For $U_0 \in \mathcal{H}$, there exists a unique  weak solution $U$ of
\eqref{e2.9} satisfying
\begin{equation}
U \in C((0,\infty);\mathcal{H}). \label{e2.11}
\end{equation}
Moreover, if $U_0 \in D(\mathcal{A})$, then
\begin{equation}
U \in C((0,\infty);D(\mathcal{A}))\cap C^1((0,\infty);\mathcal{H}). \label{e2.12}
\end{equation}
\end{theorem}

\begin{proof}
We will use the Hille-Yosida theorem. Since $\mathcal{A}$ is dissipative
and $D(\mathcal{A})$ is dense in $\mathcal{H}$, it is sufficient to show
that $\mathcal{A}$ is maximal; that is, $I -\mathcal{A}$ is surjective.
Given $G = (g_1,g_2,g_3) \in \mathcal{H}$, we prove that there exists
 $U=(u,v,z) \in D(\mathcal{A})$ satisfying $(I - \mathcal{A})U =G$ which
is equivalent to
\begin{gather}
u - v = g_1 \in H_0^1(\Omega), \label{e2.13} \\
v - au_{xx} + bv + \int_0^c F(s)z(x,1,s)  ds  = g_2 \in L^2(\Omega), \label{e2.14} \\
sz(x,\rho,s) + z_\rho(x,\rho,s) = s g_3 \in L^2(\Omega\times (0,c)). \label{e2.15}
\end{gather}
From \eqref{e2.2},\eqref{e2.3} it follows that equation \eqref{e2.15} has a unique
solution given by
\begin{equation}
z(x,\rho,s) =  v e^{-\rho s}
 + s   e^{ \rho s}\int_0^\rho e^{ \sigma s}g_3(x,\sigma,s) d\sigma.  \label{e2.16}
\end{equation}
From this and \eqref{e2.14} we obtain
\begin{equation}
(1+b)u - au_{xx} = g \in L^2(\Omega) \label{e2.17}
\end{equation}
where
$$
g= g_1 + g_2 - \int_0^cF(s) z(x,1,s)  ds .
$$
We can reformulate \eqref{e2.17} as follows
$$
\int_\Omega ((1+b)u - au_{xx}) \omega \,dx
= \int_\Omega g \omega \,dx \text{for all } \omega \in H_0^1(\Omega).
$$
Integrating by parts,
\begin{equation}
(1+b)\int_\Omega u\omega \,dx + a\int_\Omega u_{x} \omega_x \,dx
= \int_\Omega g \omega \,dx\quad \text{for all }
 \omega \in H_0^1(\Omega), \label{e2.18}
\end{equation}
that can be written as the variational problem
$$
\phi(u,\omega) = L(\omega),\quad \text{for all } \omega \in H_0^1(\Omega).
$$
By  the properties of the $H_0^1(\Omega)$, we have that  $\phi$ is
continuous and coercive. Naturally $L$ is continuous.
 Applying the Lax-Milgram Theorem, problem \eqref{e2.18} admits a unique solution
$$
u \in H^1_0(\Omega),\quad \text{for all } \omega \in H_0^1(\Omega).
$$
By elliptical regularity  \cite[Theorem 3.3.3, page 135.]{Kesavan},
it follows from \eqref{e2.17} that  $ u \in H^2(\Omega)$, and then
$$
u \in H^2(\Omega)\cap H_0^1(\Omega).
$$
Note that  from \eqref{e2.13} and \eqref{e2.16}, it implies
$ v \in H_0^1(\Omega)$ and $ z \in L(Q\times(0,c))$ respectively and
then $ (u,v,z) \in D(\mathcal{A})$. Thus the operator
$(I - \mathcal{A})$ is surjective. As consequence of the
 Hille-Yosida theorem \cite[Theorem 1.2.2, page 3]{ZL}, we have that
$\mathcal{A}$ generates a $C_0$-semigroup of contractions
$ S(t) = e^{t\mathcal{A}}$ on $\mathcal{H}$. From semigroup theory,
$ U(t)= e^{t\mathcal{A}}U_0$ is the unique solution of \eqref{e2.9}
satisfying \eqref{e2.11} and \eqref{e2.12}. The proof is complete.
\end{proof}

\section{Exponential stability}

The necessary and sufficient conditions for the exponential stability of the
 $C_0$-semigroup of contractions on a Hilbert space were obtained by
 Gearhart \cite{Gearhart} and Huang \cite{Huang} independently,
see also Pruss \cite{Pruss}. We will use the following result due to Gearhart.

\begin{theorem} \label{thm3.1}
Let $\rho(\mathcal{A})$ be the resolvent set of the operator $\mathcal{A}$
and $S(t) = e^{t\mathcal{A}}$ be the $C_0$-semigroup of contractions generated
by $\mathcal{A}$. Then, $S(t)$ is exponentially stable if and only if and only if
\begin{gather}
i\mathbb{R} = \{ i\beta :  \beta \in \mathbb{R}\} \subset \rho(\mathcal{A}) ,
 \label{e3.1} \\
\limsup_{|\beta|\to \infty}\|(i\beta I - \mathcal{A} )^{-1}\| < \infty. \label{e3.2}
\end{gather}
\end{theorem}

The main result of this manuscript is the following theorem.

\begin{theorem} \label{thm3.2}
The semigroup $S(t) = e^{t\mathcal{A}}$ generated by $\mathcal{A}$ is
exponentially stable.
\end{theorem}

\begin{proof}
It is sufficient to verify \eqref{e3.1} and \eqref{e3.2}.
If \eqref{e3.1} is not true, it means that there is  a $\beta \in \mathbb{R}$
such that $\beta \neq 0$, $ \beta$ is in the spectrum de $\mathcal{A}$.
From the compact immersion of $D(\mathcal{A})$ in $\mathcal{H}$,
 there is a vector function
$$
U = (u,v,z) \in D(\mathcal{A}),\,\,\,\text{with}\,\,\, \|U\|_{\mathcal{H}}= 1
$$
such that $ \mathcal{A}U = i\beta U $, which is equivalent to
\begin{gather}
i \beta  u - v = 0, \label{e3.3}\\
i \beta v - au_{xx} + bv + \int_0^c F(s)z(x,1,s) \, ds  = 0, \label{e3.4} \\
i \beta s z(x,\rho,s) + z_\rho(x,\rho,s) = 0. \label{e3.5}
\end{gather}
Using \eqref{e3.3} we obtain $ v_x = i \beta u_x$. Multiplying by $v_x$,
integrating and using Young's inequality we have
\[
\int_\Omega |v_x|^2\,dx
= i \beta \int_\Omega u_x v_x \,dx
\leq  - \frac{1}{2}\beta^2\int_\Omega |u_x|^2\,dx
+ \frac{1}{2}\int_\Omega |v_x|^2\,dx,
\]
from where it follows that
\begin{equation}
\frac{1}{2}\beta^2\int_\Omega |u_x|^2\,dx
+ \frac{1}{2}\int_\Omega |v_x|^2\,dx \leq 0. \label{e3.6}
\end{equation}
Applying Poincar\'e's inequality in \eqref{e3.6} we obtain
$u = v = 0$ a.e. in $L^2(\Omega)$.

Note that \eqref{e2.3} gives us
$z = v e^{-i\beta \rho s}$
as the unique solution of  \eqref{e3.5}.
Using the Euler formula for complex numbers we have
$$
z^2 = v^2[ \cos(2\beta\rho s) - i \sin(2\beta\rho s)].
$$
Taking the real part, integrating on $\Omega\times\Omega\times (0,c)$
and remember that $v=u_t(x,t)$ we obtain
$$
\int_\Omega\int_\Omega \int_0^c z^2(x,\rho,s) \,d\rho \,ds \,dx
\leq \int_\Omega\int_\Omega \int_0^c v^2 \,dx \
leq c \int_\Omega v^2 \,dx \leq 0,
$$
which implies $ z=0$ a.e. in $L^2(Q\times (0,c))$.
 But $ u=v=z=0$ is a contradiction with  $ \|U\|_\mathcal{H} = 1$
and then \eqref{e3.1} holds.

To prove \eqref{e3.2} we use contradiction argument again.
 If \eqref{e3.2} is not true, there exists a real sequence
$\beta_n$, with $\beta_n \to \infty$ and a sequence of vector functions
$ V_n \in \mathcal{H}$ that satisfies
\begin{equation*}
\frac{\| (\lambda_n I - \mathcal{A})^{-1} V_n\|_\mathcal{H}}{\|V_n\|_\mathcal{H}}
\geq n,\quad \text{where } \lambda_n = i \beta_n.
\end{equation*}
Hence
\begin{equation}
\| (\lambda_n I - \mathcal{A})^{-1} V_n\|_\mathcal{H} \geq n \|V_n\|_\mathcal{H}.
\label{e3.7}
\end{equation}
Since $ \lambda_n \in \rho(\mathcal{A})$ it follows that there exists a
unique sequence $U_n=(u_n,v_n,z_n) \in D(\mathcal{A})$ with unit norm in
 $\mathcal{H}$ such that
$$
 (\lambda_n I - \mathcal{A})^{-1} V_n = U_n.
$$
Denoting $\xi_n = \lambda_n U_n - \mathcal{A} U_n$
we have from \eqref{e3.7} that
$$
\|\xi_n\|_\mathcal{H} \leq \frac{1}{n}
$$
and then $\xi_n \to 0$ strongly in $\mathcal{H}$ as $ n\to \infty$.

Taking the inner product of $ \xi_n$ with $U_n$ we have
\begin{equation*}
\lambda_n\|U_n\|^2_\mathcal{H} -  \langle \mathcal{A} U_n, U_n\rangle_\mathcal{H}
=  \langle \xi_n, U_n\rangle_\mathcal{H}.
\end{equation*}
Using proposition \ref{prop2.1}
\begin{equation*}
\lambda_n\|U_n\|^2_\mathcal{H} +   ( b - \int_0^c F(s)ds)\int_\Omega v_n^2 \,dx
=  \langle \xi_n, U_n\rangle_\mathcal{H}
\end{equation*}
and taking the real part we have
$$
\Big( b - \int_0^c F(s)ds\Big)\int_\Omega v_n^2 \,dx
=  \operatorname{Re}\langle \xi_n, U_n\rangle_\mathcal{H}.
$$
As $U_n$ is bounded and $\xi_n \to 0$ we obtain
\begin{equation}
v_n \to 0\quad\text{as } n \to \infty. \label{e3.8}
\end{equation}
Now, for $\xi_n =(\xi^1_n,\xi^2_n,\xi^3_n)$,
$\xi_n = \lambda_n U_n - \mathcal{A} U_n$ is equivalent to
\begin{gather}
i \beta_n  u_n - v_n = \xi^1_n \to 0\quad \text{in }
 H^1_0(\Omega), \label{e3.9}\\
i \beta_n v_n - au_{n_{xx}} + bv_n + \int_0^c F(s)z_n(x,1,s)  ds
 = \xi^2_n \to 0\quad \text{in } L^2(\Omega), \label{e3.10}\\
i \beta_n s z_n(x,\rho,s) + z_{n,\rho}(x,\rho,s)
= \xi^3_n \to 0\quad \text{in } L^2(Q \times (0,c)). \label{e3.10A}
\end{gather}
By \eqref{e2.3},
\begin{equation}
z_n(x,\rho,s)  =  v_n e^{-i\beta_n \rho s}
+ s   e^{-i\beta_n \rho s}\int_0^\rho e^{i\beta_n \sigma s}\xi^3_n(x,\sigma,s)
\,d\sigma.         \label{e3.12}
\end{equation}
Using the Euler formula for complex numbers  in \eqref{e3.12} we obtain
\begin{align*}
z_n &= [\cos^2(\beta_n \rho s) -
\sin^2(\beta_n \rho s)] s\int_0^\rho\xi^3_n(x,\sigma,s)\,d\sigma  \\
&\quad -i [2\cos(\beta_n \rho s)\sin(\beta_n \rho s)]
s\int_0^\rho\xi^3_n(x,\sigma,s)\,d\sigma.
\end{align*}
Taking the real part we obtain
$$
|z_n| \leq 2 \int_0^\rho\xi^3_n(x,\sigma,s)d\sigma
$$
deducing that
\begin{equation}
z_n\to 0\quad\text{as } n \to \infty. \label{e3.12b}
\end{equation}
As an immediate consequence of \eqref{e3.12},
\begin{equation}
\int_0^\rho F(s) z_n(\rho,1,s)\,ds \to 0\quad \text{as } n \to \infty. \label{e3.13}
\end{equation}
Now we prove that $u_n \to 0$. Using \eqref{e3.9} and \eqref{e3.10} we have
\begin{equation}
-\beta^2_n - a u_{n_{xx}} = f_n(x),\quad \text{where }
f_n(x) = \xi^2_n + b \xi^1_n - \int_0^c F(s)z_n(x,1,s) \, ds. \label{e3.15}
\end{equation}
Multiplying \eqref{e3.15} by $u_n$, integrating by parts and applying
Poincar\'e's inequality, we obtain
\begin{equation*}
\frac{a}{C_p}\int_0^1 |u_n|^2 \,dx
\leq \int_0^1 (\beta_n u_n)^2\,dx + \int_0^1 f_n(x)u_n \,dx.
\end{equation*}
Writing
$$
\int_0^1 f_n(x)u_n \,dx = \int_0^1 \big[\frac{\sqrt{C_p}}{\sqrt{a}}f_n(x)\big]u_n
\big[\frac{\sqrt{a}}{\sqrt{C_p}}u_n\big]\,dx
$$
and applying Young's inequality we obtain
$$
\frac{a}{C_p}\int_0^1 |u_n|^2 \,dx
\leq \int_0^1 (\beta_n u_n)^2\,dx
+ \frac{1}{2}\frac{C_p}{a}\int_0^1 |f_n(x)|^2 \,dx
 + \frac{1}{2}\frac{a}{C_p}\int_0^1 |u_n|^2 \,dx.
$$
So, we have
\begin{equation}
 \frac{1}{2}\frac{a}{C_p}\int_0^1 |u_n|^2 \,dx
\leq \int_0^1 (\beta_n u_n)^2\,dx
+ \frac{1}{2}\frac{C_p}{a}\int_0^1 |f_n(x)|^2 \,dx. \label{e3.16}
\end{equation}
From \eqref{e3.8} and \eqref{e3.9} we have
\begin{equation}
\beta_n u_n \to 0 \label{e3.17}
\end{equation}
and by \eqref{e3.13}
\begin{equation}
f_n(x)\to 0. \label{e3.18}
\end{equation}
Using \eqref{e3.17} and \eqref{e3.18} in \eqref{e3.16}
we obtain
\begin{equation}
u_n\to 0\quad \text{as } n \to \infty. \label{e3.19}
\end{equation}
Finally, \eqref{e3.8}, \eqref{e3.13} and \eqref{e3.19} give us a
contradiction with $\|U_n\|_{\mathcal{H}}=1$. The proof is complete.
\end{proof}


\subsection*{Acknowledgments}
The authors would like to thank the anonymous referees for the careful reading
of this paper and for the valuable suggestions to improve the paper.
This project was partially supported by UFBA/CAPES (Grant No. 008898340001-08)
and LNCC/CNPq (Grant No. 402689/2012-7).


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\end{document}